In a blender, whirl papaya, 1/2 cup sugar, and 1 cup water until very smooth. Pour into a large pitcher (at least 3 qt.). Whisk in 1/3 cup lime juice, 7 cups water, and more sugar and lime juice to taste. Chill until cold, at least 1 hour or up to 1 day. Serve over ice.
Nutritional analysis per cup.